This Gun for Hire (1942) [Philip Raven]: Shot in the back by police while trying to escape out the window; he dies shortly afterwards while talking to Veronica Lake.

China(1943) [Mr. Jones]: Reportedly killed in combat with Japanese soldiers. (I haven't seen this movie myself, but got the information from a review.)

Shane (1953) [Shane]: Arguably mortally wounded in a shoot-out with John Dierkes; some viewers have argued that Alan is already dead in the final shot of him riding out on his horse, while others believe that he survives the movie. (The 1998 film The Negotiator includes a sequence of Samuel L. Jackson and Kevin Spacey debating this very point.)